- FindStat

Identifier

Mp00064: Permutations —reverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤ

Values

[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[1,3,2] => [2,3,1] => [3,1,2] => [3,1,2] => 0
[2,1,3] => [3,1,2] => [2,3,1] => [2,3,1] => 1
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [3,4,2,1] => [4,3,1,2] => [4,3,1,2] => 0
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 1
[1,3,4,2] => [2,4,3,1] => [4,1,3,2] => [4,1,3,2] => 0
[1,4,2,3] => [3,2,4,1] => [4,2,1,3] => [4,2,1,3] => 0
[1,4,3,2] => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 0
[2,1,3,4] => [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 1
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 1
[2,3,1,4] => [4,1,3,2] => [2,4,3,1] => [2,4,3,1] => 2
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 1
[2,4,3,1] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 0
[3,1,2,4] => [4,2,1,3] => [3,2,4,1] => [3,2,4,1] => 1
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 1
[3,2,1,4] => [4,1,2,3] => [2,3,4,1] => [2,3,4,1] => 2
[3,2,4,1] => [1,4,2,3] => [1,3,4,2] => [1,3,4,2] => 1
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[4,1,3,2] => [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 0
[4,2,1,3] => [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[2,3,5,4,1] => [1,4,5,3,2] => [1,5,4,2,3] => [1,5,4,2,3] => 0
[2,4,3,5,1] => [1,5,3,4,2] => [1,5,3,4,2] => [1,5,3,4,2] => 1
[2,4,5,3,1] => [1,3,5,4,2] => [1,5,2,4,3] => [1,5,2,4,3] => 0
[2,5,3,4,1] => [1,4,3,5,2] => [1,5,3,2,4] => [1,5,3,2,4] => 0
[2,5,4,3,1] => [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[3,2,4,5,1] => [1,5,4,2,3] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[3,2,5,4,1] => [1,4,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 1
[3,4,2,5,1] => [1,5,2,4,3] => [1,3,5,4,2] => [1,3,5,4,2] => 2
[3,4,5,2,1] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[3,5,2,4,1] => [1,4,2,5,3] => [1,3,5,2,4] => [1,3,5,2,4] => 1
[3,5,4,2,1] => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[4,2,3,5,1] => [1,5,3,2,4] => [1,4,3,5,2] => [1,4,3,5,2] => 1
[4,2,5,3,1] => [1,3,5,2,4] => [1,4,2,5,3] => [1,4,2,5,3] => 1
[4,3,2,5,1] => [1,5,2,3,4] => [1,3,4,5,2] => [1,3,4,5,2] => 2
[4,3,5,2,1] => [1,2,5,3,4] => [1,2,4,5,3] => [1,2,4,5,3] => 1
[4,5,2,3,1] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[4,5,3,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[5,2,3,4,1] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[5,2,4,3,1] => [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[5,3,2,4,1] => [1,4,2,3,5] => [1,3,4,2,5] => [1,3,4,2,5] => 1
[5,3,4,2,1] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[5,4,2,3,1] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 5 + q$

$F_{4} = 14 + 8\ q + 2\ q^{2}$

Description

The number of occurrences of a type-B 231 pattern in a signed permutation.
For a signed permutation

$\pi\in\mathfrak H_n$ , a triple

$-n \leq i < j < k\leq n$ is an occurrence of the type-B

$231$ pattern, if

$1 \leq j < k$ ,

$\pi(i) < \pi(j)$ and

$\pi(i)$ is one larger than

$\pi(k)$ , i.e.,

$\pi(i) = \pi(k) + 1$ if

$\pi(k) \neq -1$ and

$\pi(i) = 1$ otherwise.

Map

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation

$\sigma$ of length

$n$ is given by

$\tau$ with

$\tau(i) = \sigma(n+1-i)$ .

Map

to signed permutation

Description

The signed permutation with all signs positive.

Map

inverse

Description

Sends a permutation to its inverse.